asymptotic expansions of hypergeometnc functions

^>.1

XIV. Asymptotic expansions of hypergeometnc functions.

By G. N. Watson, Sc.D., Trinity College, Cambridge.

[Received June 11, 1917. Read Feb. 4, 191S.]

1. The hypergeometric function F{a,j3; y, x) presents two distinct problems to mathe-maticians interested in the theories of analytic continuations and asymptotic expansions. Thefirst, and simpler, problem is that of finding the analytic continuation of the function beyond

the circle x = 1 , which is the circle of convergence of the series by which the function isusually defined, ilore generally, the problem is that of finding the analytic continuation of

q+iFq beyond the circle | , = 1 , and of finding the asymptotic expansion of ^F, for large values

of j a; I when p

278 Dr WATSON, ASYMPTOTIC EXPAXSIOXS

More modem investigatdons in the particular case of Legendre functions are due to Barnes*,but his methods convey the impression that they are primarily adapted for attacking the first

problem rather than the second; on the other hand, it must be stated that the}" are quite

effective in obtaining complete asymptotic expansions of P,l^{z) and Q'"(^) when either

I

n or j m 1 is large, and n, m are not restricted to be integers.

The most natural way of attacking the second problem seems to me to be by the method

of steepest descents. The application of this method to the problem is of special historicalinterest, because it was in connexion with hypergeometric functions that Eiemann ^vrote thepaper-)- which contains the first indications of the potentialities of the method. More recently,

in the hands of Debye, Brillouin and myself, the method has proved to be effective in dealing

with Bessel functions, Weber-Hermite functions (i.e. those associated in harmonic analysis with

the parabolic cylinder) and numerous other functions (defined b\' definite integiitls of various

types) which occur in many branches of Mathematical Physics*

As the investigations of this paper have the asymptotic expansions of Legendre functions

as one of their ultimate objects, a notation will be employed which will make it as easy aspossible to write do\vn the special results for these ftmctions. It may be mentioned here that

the contours which are yielded by the method of steepest descents in the case of h\-per-

geometric functions are all algebraic curves, many of them being nodal circular cubics ; this is in

marked contradistinction to the fact that the contours employed in pre\"ious applications of the

method (with the exception of some of Brillouin's researches on the functions of Physical Optics)

have been somewhat complicated transcendental curves.

It should also be remarked that very slight modifications of the contour integrals are

adequate to supply the asymptotic expansions in various exceptional cases in which the appli-

cation of the Mellin-Bames method requires detailed separate investigations.

* Quarterly Journal, loc. cit. This paper contains ^x

OF HYPERGEOMETRTC FryCTTOys. 279

Pabt I. The geseraused Jacobi-Tchebychef fcxctions.

2. Statement of the problem.

In this section, we shall obtain the complete aspnptotic esp\nsion .t

F{a+ K ^-^; 7; >.where a, ^. y. x have any assigned values, real or complex (the value of , being not restricted

to be less than 1), and X, is large while* argX

280 Dk WATSON, ASYMPTOTIC EXPANSIONS

siTg(zt) is defined by giving it the value arg(a + l) at ^ = 1. The integral obviouslyconverges at both limits when X , is sufficiently large.

To evaluate the integral, suppose that | 1'>2; and then, taking the path of inte-gration to be the real axis, we have 1 ^ < \zl\. Hence, on expansion, we get

A = (." -D f ' (I - o-*-> (1 + ?)>-' I ^!V/^"? f?-^ fdt.J -1 =o r(a-|-\) \I -ziIntegrating term by term, we have at once

^2'^-^T(a + \-y+l)r(y-^-i-X):z-l\-'-^f

^ ^ ^ ^ .. .2 \

.

^ = Tia-l^2X+{) (V) i^(+X,a-fX-7-^l:a-^+2X+I:^--J.Since /, is anal}i;ic and one-valued throughout the plane (when cut from + 1 to x ), thisequation, proved when \zl\>2, persists throughout the cut plane.

Xext take the integral /,, defined by the equation

/., =1 (1 - ty-^-y ( 1 +

OF HYPERGEOMETRIC FUXCTIONS. 281

Then

1, 1,-j^^ ,u.{. l)e , -r ,-^,,,Y{^-X~y + \) 2"X {(1 - u){z - l)}--^ {z -\)du

"' ^"r(2-7) I 2 ;

X f (a + X-7+1, /i-X-7+ 1; 2-7; i - U)_27rie'"V2^-^r(a + \-7+l) /s- 1 ,'-v,

^ , o ^ ,i o 1 i n^

r(2-7)r(a + \)Since it follows from the formulae of analytic continuation* that

2^-^r(7-^ + x)r(i-7)^>=

r(i-^ + x) i^( + ^, /5-x, 7, .-i^)

^2--^r(a + X-74.1,r.7-l) ^^:zJ)-V(.X-7.1, ^-X-7 + l: 2-,: i - *.),

it is easily shewn that

strictly speaking, this result has not been proved when 7 is a positive integer, in view of

the factor r(l-7) which occurs in the course of the reduction; but, since both sides ofthe final equation are analytic functions of 7 (except when 7 is zero or a negative integer),

the equation holds also for the exceptional values of 1, 2, ... of 7, provided only that X ' is

so large that the integrals /i and I2 are convergent.

4. The contours provided by the method of steepest descents.

We now- apply the principles of the method of steepest descents to the integral

((1 - t)''-y ( 1 + t)y--' (z - 0"" exp (- X log ^,j dt,

with a view to determining asymptotic expansions of /, , I., whenJ

X ) is large.

The stationary points of log {(2 - 0/(1 - t')],

282 Dr WATSON, ASYMPTOTIC EXPANSIONS

( t-z^ _F(Z=4-r-+l)-sinhfsin77(Z-- F l) + 2XFcosh^cosj7Since tan

| / log^-j^^J

= x (Z^' + F' - 1) - cosh ? cos v (X' - T' - 1) - 2A' Fsinh ^ sin rj'

the equations of the two cubics are

Y{X- + Y- + 1) + sinh f sin r) {X- - 1"- - 1) - 2Z F cosh f cos t;= tan7;[A'(A'-+ F'-- D- cosh f ccs 7; (X- - F- - 1) - 2A'Fsinh f sinT;].

and the cubics have nodes at (e^ cost?, e^ sin?;) and (e^f cos j;, - g-^ sin ijU-espectiveh".

We shall consider fully ( 5-7) the properties of the cubic (S.^ obtained by taking the

upper sign in this equation, and deduce (

OF HYPERGEOMETRTC FUNCTIONS. 283

It is easy to see ti-Diii the table that {when ij is a positive acute angle) as ^ increasest'roiii tan y to + x , and then from x to tan 77 again, it passes in succession thi-ough tlie vahies*

Ml' M-'. (Ms '>!' Mj). M.-,. Ms. Mi-

To determine the positions of the node relative to these six points, we observe that theparameters of the node are the roots of the quadratic g (n) = 0, where

g (yfj.) = fjL- e^ cos -qhrnri -\- fj, (e^ cos Irj e~^) e^ sin >; cos 77

;

now g (fi,) >0>g (/x.,),and g (/n..) = 2e~^ sin 7? (cosh f cos i])/{e^ cos 7; 1 )- > 0,

g (yti,) = e~^ tan 77 < 0,

g (tj,2) = e~^^sin 7; cos 77 (coth- f tan- 77 + 1) > 0.

Hence one of the parameters of the node lies in the intervals (/x,, /is) and (/^j, ^(g), whili' the

other lies in the interval (/^i, /i,).

Hence, if a point starts fi-om infinity and traverses the entire length of the cubic, it

passes through the points of interest in the following order:

cK, ef{node), z, 1 or e^ cos 77, ef(node), ie^sin77, 1, 00

(the points being specified by their complex ciwrdinates); if 77 is a negative acute angle, the

points are traversed in the same order; while if 77 is obtuse (either positive or negative), the

order is:X, e^ (node), z, 1 or e* cos 77, e^(node), te^sin77, 1, x.

The curve is shewn in Figures 2 and 3 in two cases, 77 being a positive acute angle, and

e^ cos 77 > 1 in 2 and e^ cos 77 < 1 in 3, while in Figure 4 77 = |7r. The portions of the curve from

Fig. 2. Fio. 3. Fig. 4.

which the contour has to be selected are shewn in continuous lines ; and it is obvious (even inthe critical cases when 77 = + Jtt) that the cubic has an arc which passes from 1 to 1 throughthe node without passing through f the critical points z and 00 ; and further, this arc cannot

* The sign of e^ cos rj - 1 determines whether pL^ comes is true if cosech | cos rj < coth J, which is obviously the case,before or after /ii4

.When e^ cos ))>1, wc prove that jn^P"^^- t This is not true in the other critical cases 7; = 0, jr;

cedes ^3 by proving that coth f {e^ cos 7; - 1)


be reconciled with the segment (-1, 1) of the real axis without passing over the point z, and

the are is of the type specified for the second contour of 3.

Further, as t passes along the arc from 1 to 1,

T = log l(f - ^)/(f l)i + log ( 2eO

passes by a steady decrease from + x to (at i = e^) and then increases steadily to + x again.

Hence /, = ([" + [")(!- tY'^ (1 + i)y-^-'(z - t)-^ 2^ e'^ e"^' J^ dr.

6. Transformations of the integrand of /.,.

It is now possible to obtain the asymptotic expansion of /._,, by a consideration of the

expansions of(1 - tY-y(l + t)y-^-\z - O"" {dtldr),

in powers of t on the two parts of the path near t= e^.

From the equation defining t, it is readily seen that

; _ ef = + (1 - f^4-')i (1 -e-')- - ef (1 - e-^)

= + i a,T''+i+ i hr^\

when |T is sutHciently small; if the upper sign refers to the arc joining ^ = e^ to t=\, then

a = + (l-e=0*,

where the upper sign is given on the understanding that a^ varies continuously with ^, so

long as /(eO '^o^* not change sign, and Oq is positive when 7; = + Jtt, so that the saddle-point {i.e. the node of the cubic) is on the imaginary axis in the

OF HYPERGEOMETRIC FUNCTIONS. 285

Therefore C = 2-' e? ( 1 - eO* " "^ ( 1 + e^y - ^^ - (^ - i

.

We shall next obtain a compact expression for the general coefficient ci, it is evident that

I rtO+,()+) ( rlt) dr

the path of integration being a double circuit round the origin in the r-plane corresponding to

a single circuit in the f-plane round t = e^.

Hence Ccg is half the coefficient of (t ef)-' in the expansion of

(l-tY-y(l + t)y-^-'(2~tr''T-'-Ht-e^f'+'in ascending powers of t eK

If we write i = e*"+ T, it is easy to shew that c^ is e(jual to (1 e-O" multiplied by the coef-

ficient of jr-' in the expansion of

in ascending powers of T.

The value of c,j is unity, as has been already stated ; and

c, = i (X + Me^+ Ne-

'286 Dr WATSON, ASYMPTOTIC EXPANSIONS

7. The asymptotic expansion of I.,.

It follows, by applying a general theorem* to the result of 6, that we have the com-plete asymptotic expansion

i a-t)'^-ya + ty--'(z-t)-'e-^-'{dt;dr)dTs,G S cA T'-ie-^uh+ ^ dj T'^-^-rfr,and so /, ^ -2*+" Ce^i i c,. p (s + i) \ - " 4

,

the expansion being asymptotic in the sense of Poincare whenj \ , is sufficiently large and

l^rgXj ^^TT-^B: provided that rj is not equal to or +7r, as the integrand would then havea singularity for a positive value of t.

We shall now examine to what extent these restrictions can be removed.It can be shewn that the expansion is valid when 7/ = or + tt; for suppose that tj is slightly

greater than (or - tt); then instead of taking the contour to be the real axis in the r-plane,we take it to bef the ray argT = - iS; the modified integi-al is an analytic function of \ when- ^TT + S < arg \ < ^TT, and it is also an analytic function of ? when /(?) 5^ (or - tt). Hence,making f assume the real value '^ (or the value f - iri), we see that, when ?? = + or - tt + 0,/ is equal to the modified integral, and also the asymptotic expansion is unaffected. To discussthe cases rj = 0, tt -0 we proceed similarly, but we swing the contour round in the oppositedirection; and we note that the expansion is the same whether j; = + or - 0.

Secondly, to extend the range of values of arg X, we observe that the process of swinginground the contour can be can-ied further, as shewn in Fig. .5. Take the two of the points 2f + 2A-7ri

Ki.i,

Proc. Lonliin Math. Soc. (2) xvil. (1018), p. 133. deformed contour, is oue which passes from - 1 to 1 andt The contour in the f-plane, corresponding to this it is of a spiral form near eacli of these points.


which are nearest to the real axis (one on each side of it), and let the rays joining the originto them be arg t w,, arg t = w^, so that tui, w^ are positive (or zero) acute angles.

When argX,>0, we take the contour to be the ray argT = tuj + ^8, and the modifiedintegral (which has the asymptotic expansion already given) provides the analytic continuationof / over the range for which arg A. < Jtt 4-


The parameters of the node are the roots of h {/j.) = 0, where

h(fi,) = fjL- e~^ cos Tj sin rj fi, (e"- cos 2j; e^) e~^ cos tj sin 7;

;

now k{fu)^^e-''''^dT,.

Also, when t is e~f we have

l-t=l-e-


There is, however, an important difference in the range of values of arg X, for which the

asymptotic expansion of I^ is valid. For the singularities i)f the integrand are the points

T] = ikiri, 2kiri 2^, and the points 2kTri 2f, being on the left of the imaginary axis, do nothamper the process of swinging round the path of integration; we may therefore swing it roundso as to be either of the lines argxi = + (^tt iS), according as 7(\) g 0; and therefore theasymptotic expansion of /, is valid over the sector

I

arg XI

^ TT S,

provided that|A,

jis sufficiently large. For brevity we shall describe the sector for which

I

arg X!^ TT S,

as a complete range of values of arg X, while the sector for which argX lies between lirw.+hand \tt + ft), 8 will be described as an incomplete range of values of argX.

9. Asymptotic expansions of hypergeometric functions.

It is now possible to write down the asymptotic expansions of two independent solutions of

the h\'pergeometric equation of the type under consideration; the formulae are as follows:

- IX"""'^ / 2F(a + \, a + X-7+1; a-j3 + 2\ + l; -_

j

2-^r(-/3 + 2x + i)._,,.^i_^_^i-,(i + ^-,).-.-^-4 i e;r(. + i)x 4,

r(a + X-7 + l)r(7-/3 + X)valid when | X j is large and | arg X tt 8 : and

^( + X, ;3-X; 7: i- i.)^ ^


Part II. Asymptotic expaxsioxs of Legendre functions.

10. The asymptotic expansion of Qn {s) ivhen \n\ is large.

As special cases of the formulae obtained in | 7-9 we can write down cninplete

asymptotic expansions of the generalised Legendre functions Q,(2), P"'(z) when z is assigned

and either iii or |rK| is large, n and m not being necessai-ily integers; the formulae agi'ee with

results previously obtained by Hobson and Barnes, but some of them are valid o^er a more

e.xtended range than has been hitherto assigned.

Let us first consider Q""(2) when n\ is large and arg n ^tt-S. We have*

sin( + m)7r U-1/ r(2K+2) 12 V /< \^ j_^/when iarg(2+ I) ^-tt, so that

z-l~\l-e-i) '

the arguments of both sides of this equation being equal and not differing by a multiple of 2ir.

But, with these conventions, if a = 1, /3 = 0, 7 = 1 - m, X = n, we have

Therefore, by 8, the asymptotic expansion of Q,,*" (z) is given by the formula

n mi ^ -rCi + l) sin (n + m)ir (7r/)-e-"'+"^ [. , | c^T(+_|)|^" ^^^"'r(n-m+l) sinn-n- V(l-e-^0 1 s=i r(i)- \'

valid when ,arg(2 + 1) j ^tt and argw | 7r 8; and c,' is given by the formula

c, = = ?((- J + ( III- i) coth f.4(l-e-'-f)

In the special case when z = cos 17, Qn'" (z) is defined t as

i {Qn" (cosh (0+ir,)) + Qn"' cosh (0 - ir])},

where we may take < ?; < tt.

If we wTite in turn ^=0 + iv and ^ = - ir). we gete< = e-'^ V( 1 - e--^) = e = ^''' = l""'' V(2 sin rj),

since V(l e~^) is positive when ^ is a pure imaginary.

We thus obtain the complete asymptotic expansion

^ ^ '' r(n-TO + l) sm NTT V \2wsinj?y L i ^"(*" i)cot n . , , ,

,

11,-^ * ^'sin{( + .U; + i7r! +

valid when < ?; < tt, jarg nj < tt 8.

This 18 in accordance with the definition given by (p. U4). It differs from Hobson's definition, Phil. Tnnn.

Barnes, Quarterli/ .Journnl (loc. cit.). pp. lOO, 107. ('"c cit.). p. 471.

t This is also in accordance wit)i Barnes' definition

OF HYPRRGEOMETKIC FUNCTIONS. 291

11. Tlie


If in the first of these formulae we wi'ite m. for m, and then in the fommlae of | 3we write a = /3 = i?(+J, 7 = + |, X = \m, we see that

9-fm+ Jn + i/ ;_i\-Jn-i" -^ ' r(Jw-in)r(im + | + l) "

where, in I^ we have to Avrite (1 s'-)~' in place of \{\ z) wherever z occurs; so that f is^now defined in terms of z by the equation

i (1 - cosh = (1 -')"'.

and therefore cosh^=^=-, sinht'=^^i,2^1 Z' L

and, since R{z) ^0, the upper sign must be taken in order that we may have .e~^\ ^\.

The asymptotic expansion of P~"'(2) is therefore

where c/ (1 + 2P(42)~* is the coefficient of T-* in the expansion of

r(i+g)- ( {i + zfT-- 11"'"*'^l4:zT' ^^\ ^ ^s-2{z^-\)T-(\+zfT^\\

and the asymptotic expansion is valid, when jR(2)^0, over a complete range ( 8) of valuesof arg m.

By using the second asymptotic expansion given in | 9, we find

-^ U + li r+,-:"T^|)7n )J-this is valid, when iJ(2)>0, over an incomplete range of values of a.rgr. the upper or lower

sign is taken according as / (-J $ 0, i.e. as I(z) g 0, and c* is derived tiom c,' by changing

the sign of z.

Frnin the formula (Hobson, p. 462; Barnes, p. 109)

2Q"' (2) r ( - >u - n ) 7r-= ,sin m-jr sin n-rr =--f""-'M- - Jf^'] . ,

we find P,r (z) -.,- .v^.^^..^2;xv^.-.,.-r^; ,

_^.^ ^^^^ / iZli) "

'" i 1 + f2* c. r (* + J )\find P'"(2)-, !^ ?

^-^j^^^

= ^' smwTT = U T^ - ^,,. ,

This result is simplified by the disappearance of the first series in the special case wlien

7/1 is a positive integer.

The general formula is valid, when R (z) ^ 0, over an incomplete range of values of arg di ;the special formula is true (jver a complete range.


Since $' {z) F (- m - n) = Q,r" (2) T {m - n)

(Barnes, p. 105), the asymptotic expansions wht'o B{z)'^0 are completed.

We next consider the expansions when R {2} ^ 0.From the formula (Hobson, p. 463 ; Barnes, p. 106)

P, {-z) = P,.'" {z) 6=^'""' - 2Q,.'" (z) TT-' sin mr,we see that, when R (z) ^ 0,

z-l\i'"(, , ^ 2%T(s + Dsin WITT T i 1 +

z+lj ] \Zi r(i)m'-fz + l\i"^( - 2'c,r(s+i)

- sni mr r -^ 1 + i

_

2"'r(im + + |)r(|m-l?t+i)^^'^ 7rv/(2m7r)

Kz-iJ 1 " ,=1 r(i)mChanging the sign of z, and noting that, when this is done, Cg interchanges with Cg' and e"""with e~""", we see that the expansion of Pn"-{z) has the same form for all values of z in an

incomplete range of values of arg in.

Next, taking the formula (Hobson, p. 463; Barnes, p. 106)

Q.'"(-^) =-e*""Qn"'(^).

we obtain the asymptotic expansion of Q"'( 2) when R{z)'^0; writing z for z, we see that,as in the case of P,i"'(a), the asymptotic expansion of Qn^iz) has the same form whether R{z)be positive or negative; and from formulae already given connecting Pn^, Qn^^, it is evident

that the same is true of Pn~{z) and Q,~^"-{z). The expansion of P~'"(2) is valid for a completerange of values of arg hi when R{z)^0 only; the other exj)ansions are valid for an incompleterange, as is that of P,,"'" {z) when R {z) ^ 0.

The permanence of the form of the expansions when R{z) passes through zero might havebeen anticipated* from the fact that, when m is positive, (^ + 1)'" and {z 1)'" have the samemodulus when R{z) = 0, and that (as was pointed out by Stokes) discontinuities in asymptoticformulae usually occur in terms which ai'e negligible in comparison with the dominant part of

the expansion.

Part III. Miscellaxeous properties of Legendre functions.

13. Definite integrals representing Legendre functions.

We shall now obtain important and interesting definite integrals for P (cosh f) andQn (cosh f) ; they are derived fi-om the formulae of 3, 6, 8 by putting a.= 1, ^ 0, 7=1, \ = n;this substituiion gives

/.-,= ('/" +I ]

(cosh ^- t)-' 2"e"ie-"-'{clt/dT) dr,

, ,* cosh t

, .

where t = log ~ + low 2eS

so thatdr__ (t--l) (f -cosh^dt~ i + l, -)i; m+1; i-fe),from the fact that P,~'" (z) is expressible in terms of which is of the form described in Part IV below as type B.

Vol. XXII. No. XIV. 38


Let ij.fs be the values of t corresponding to any assigned positive value of t. of which fj is

on that part of the contour S^ in the i-plane which joins the node to the point t= + \.

Then, since t^, ,

'0 (i_e-'-)*(l_e-'?-r)*

pro\4ded * that 77 is not + tt.

It is desirable to modify these results slightly, by observing that

1 - e=f-^ = e^ { - (1 + e-') sinh ?+ (1 - e"') cosh ?}= e^f^e^ {(1 + e-') sinh f- (1 - e"') cosh f},

according as 7(5') S 0, where arg (sinh f) lies between + tt.

Similarly1 - e-=f-' = e-f {(1 + e-'') sinh r+ ( 1 - e"') cosh ?}

;

and so we have the formulae

/, = 2+'e-


When z is on the real axis, however, we define Q (cos rj) as the mean of the values on eitherside of the cut ; so that

Q (cos 7) ) = |Q {cosh (0 + %r])\ + |Q [cosh (0 - it])] ;since sinh (0 + t?;) = e* -'^' sin t} when < ?; < tt, we see that, for values of t) between and tt, luehave

r^ p-{n+VTQ (cos t;) = A e

196 DrWATSOX, asymptotic EXPANSIONS

We may consequently write

^ {(I + e-' ) sin t>î{l- e"') cos 0}" ^ rfr = iw Fe-*

where W is positive and w is an angle (depending on n and 0) between and jtt i0.We thus obtain the formulae

P (cos d) = Fcos {{n + i) 0-1-77 + co], Qn (cos 6) = ^ttIFcos {(n + ^)d + l-7r + m}.

Next it will be shewn that (n + ^) 6 + co is an increasing function of when n is fixed.

We have ttP,, (cos 6) ., ,.,-,,,

and so

1 {(n + J) ^ + iTT + 0,} = 2,r |q (cos 6) dPnicose) _ ^^^ ^^^^ ^^ dQ.,ico,0)y j"^, ^^^ ^^^^ ^^j,+ 4{Q,.(cos^)j

But, by appl3"ing a well-knowoi theorem, due to Abel*, to Legendre's equation, we deduce

that sin- [QPn PnQn} is constant; and, on wTiting 6 = iTr and making use of the values ofP(0), P'(0), QniO), Qn'iO) given by Hobson, p. 469, and Barnes, pp. 121. 12-1., we get

sm'd{QP,:-P,Q,:] = -l.

Therefore ^ {{n + i) +> + } = 2 {tt sin ^W] > 0,which gives the result stated.

We can now obtain limits for the zeros of P{cosd)-. when O^d^W, we observe that(n + ^)d l'7r + a) certainly lies between {k J) tt and {k + J) tt, k being an integer, if

(fc-i)7r$(n + i)6l-i7r, ( + ^)(9-i7r + (i7r-i6l)$(i + i)7r,. (4^-1 )77 ^^ (2^+l)7î.e. if i

J, $ c' ^ s4;i + 2 2>iIn this range of values of 6, cos {( + h)6 Iw + co} has the sign of (1)*; therefore, as

increases from (2^ + l)7r/(2H) to (4i + 3)7r/(4n + 2), (n + h)0 ^tt + o} steadily increases froma value between"(^- + })7r to a value between (k+l + |)ir; hence its cosine changes sign onceand only once. Thus the only zeros of P(cos^) in the range O-^^^stt are in the intervals

[{2k + 1) 7r/(2n), (4A + 3) 7r/(4i + 2)]

;

and there is one zero and only one in each of these intervalsf-

When \v^d^7r, co is negative, so that the inequalities are replaced by{k-^)-7r^{n + l)0-l0,{n + l)0-}-7r^{k + i)7r:

and we get one zero and only one in each of the intervals

[{ik + 3) 7r/(4 + 2), {2k +l)ir {2n)]

and none outside these intervals.

The function Q (cos 0) can be dealt with in a similar manner ; we shall not give the detailsas the reader will have no difficulty in constnicting the analysis.

Crelle n. p. 22. See also Forsyth, Differential Equa- internal point unless n is an odd intecier. in which case thetionn, 6s. The dashes denote differentiations with regard corre.'iponding interval is evanescent, and we have the knownto cos . result that P (0)= 0.

^ None of these intervals can have the point îr as an

OF HYPERGEOMETRIC FUNCTIONS. 2'J7

Part IV. Asymptotic expansions of a system ok hypergeoi^etric functions.

15. The system of hypergeometric functions with large constant elements.

We shall now determine the asymptotic expansion of any hypergeometric function in wliichone or more of the constant elements is large, provided that, when more than one of the constantsis large, the ratio of the large constants is approximately + 1. The Jacobi-Tchebychef functionsdiscussed in Part I are the most obviously important functions of this nature, but others seem tobe of sufficient interest to justify the very brief account which we shall give.

The functi(3ns which will be considered are of the form

F (a +^\, ,8 + .^\

; y + e^X; x),

where a, /3, 7, a; are assigned, j \ | is large and e,, e,, e^ have the values 0, + 1.

There are obviously 27 sets of values of (e^, eo,3), but of course the set (0, 0, 0) has to be

omitted ; and nine other sets may also be omitted on account of the symmetry of the hyper-geometric function in its first two elements ; thus (1,-1, 1) is effectively equivalent to ( 1, 1, 1).

We shall take the hypergeometric equations associated with the surviving seventeen functionsand obtain asymptotic expansions of a fundamental pair of solutions of each equation. It willappear that the equations tall into four distinct types, according to the values of (e,, e.,, e,^ shewnin the following table :

Case

298 Dr watsox, asymptotic expansions

16. Hypergeometric functidns of type A.The reader will have .observed that the functions of type A are those already investigated in

Part I of this paper, in^-iew of the fact that the function of case 1 is jP(a + \, /3 \; 7; x). Weshall merely give a table, indicating the natiu-e of the order of magnitude of the constant elements

in the twenty-four h\-pergeometric functions connected with the equation which is satisfied by

F(a + \, /3 X; 7; x). By expressing an}- one of these functions in terms of the two funda-mental integrals /, and Z, introduced in Part I, the asymptotic expansions of the twenty-four

solutions are at once obtained for a range of values of arg\ greater than the half-plane !argX|$|7r;for values of \ outside this range, we put X = X,, and then we obtain the asymptotic expansionof the function under consideration in terms of Xj.

The complete set of functions of type A is given in the following table, the numbering ofthe solutions t)eing that adopted by Forsyth, Differential Equations, 120-121 ; the first three

columns in the table give the coeflScients of X in tlie corresponding elements of the hypergeometi-ic

functions connected with the solutions shewn in the fourth column.

Coefficients of X 1 Functions Case


The two solutions which will be regarded as fundamental are the sulutioii (I), namelyFia, /3 ; 7 + X ; ), and (VIII), namely a;i->-* {l-a:)y+'^-'-^F(l - aA- /3 : y + \- a- (B + 1 : 1 -x);it is evidently sufficient (since both are included in case 4) to obtain the asymptotic expansion of

one of these functions.

The reader will easily verify that these solutions form a fundamental system when| X j is

large.

We now investigate the asymptotic expansion of F{a, /3 ; y + \; .i) ; it will be found that,in the case of this function, the saddle-point, which is usually characteristic of the method ofsteepest descents, does not appear in the analysis.

We take the integral* L,= [ t^~'(l - t) r+^-^-i (l _ j;t)-''dt,

and we observe that (when X, is positive), (lt)'' decreases steadily from 1 to zero as t describesthe path of integration.

Accordingly, writing 1 ^ = e^', we have

/s = f {(1- e-')S->e-"r-si {1-X + xe-')-'} e'^'dr.

Jo

Now, when t is sufficiently small, we have

(1 - e~^y-'e-^'y-^> (1 - + xe-^)-" = t^-' S k^T',.5 =

where k^, = 1. Hence, when R (^) > 0, we have

/3~ir(/9 + s)^,/X^+;s =

the singularities of the integrand are at the points t = 2mri, 2mri + log (1 - x'^), and so, as in 8,the expansion is valid over a complete range of values of argX when 1 1 x~^\ ^ 1, and over acertain incomplete range (greater than a half-plane) when 1 1 ~'

|> 1.

Hence ^(a, /3 ; 7 + X ; ^0 -^ ^^~^'^. i ^^^^^ ' ' r(7 + x-/S)x^=o r{^)\^When R{/3)^0 this result may be obtained by taking a Hankel-Pochhammer contour (x ; -I-)in the r-plane in place of the real axis.

The expansion on the right may be obtained formally by taking each term of the series for-'^(O) ^; 7 + '^; ^)! expanding in descending powers of X, arranging the sum in powers of X, anilmultiplying by the expansion of T (7 -1- X - /3)/T (7 + \) in descending powers of \.

18. Hypergeometric functions of type C.

This type consists of the twenty-four functions associated with the equation for whichsolution (I) is the function F (a, 13 + \; y-\; *); the coefficients of X in the constant elementsof the twenty-four functions are as follows :

* If .r>l, an indentation has to be made at t=\jx in the path of integration.


Coefficients of X


It is found that the only finite singularities of t qua function of t ai-e at the points in the r-plane

for which t = l; these are the points t = 2.S7ri. The finite singularities of the other function

are at the points in the r-plane at which t = lix: these are the points t = 2s-7ri log,2sTri.

One of these points is on the real axis ifi

. = 1 or if x is negative ; and one of the points

approaches the origin if, and only if, a; ^ 1 ; hence the expansion holds for an incomplete rangeof values of arg \ except at x= l; and it holds for a complete range of values of arg \ when4.r >ja;-l \\

The domain of the complex variable ./ for which this inequality holds is shaded* in Figure 7.

Fig. 7.

When X = 1 the expansion assumes a different form, since ( 1 xt)~'^, when expanded inascending powere of t, has its leading term t"-" instead of (1 -t-a-)""; it is easy to make the

necessary modifications in the analysis.

Next take the integral

/, = x'-y+>' (1 - x)y-'-^-"'i {- 0"^""^ (1 - tf-^^-^ (1 - .rt)"-' dt.

The method of steepest descents gives the same potential contours as in the case of I^, but

now, in order to secure convergence, we take the circle t =1 (taken counter-clockwise startingfrom i= 1) as contour ; if a; < 1, by expanding in ascending powers of x, we get

If1a; ' > 1, however, by expanding in descending powers of x, we get

/, = x^-y^^ (1 - a;)v--3-^ l(_ i)-s-i-A ^j _ tf-y+->^ (^ ~ l) '^^

r(l+ 13-y + 2\)

= - 2-n-i af'-y-'^ (I - a')v-

30i Dk WATSON, ASYMPTOTIC EXPANSIONS

To obtain the asymptotic expansion of 7^, we write

(l_f)=/(_40 = e-,

and then, if gi is the coeflScient of T-" in the expansion of

xT Y-' (4 . ,, . r- M-*-i

we get

(-y^a - TY-^ (1 - ^Tf-r [l - j^J"' 1^, log ^1 +^)/, ^ - /2-i'+'-""-^ (1 + a)"-' A'^-r-^ (1 _ .,,)r-


The path for I-, is reconcilable with the real axis without crossing over the singularity ^3= Yjx;the path tor / passes above or below this point according as the point is above or below thereal axis.

It is readily verified that

^^=r(7 + 3X) PKc^-^K^ + X: y + SX: .r),

xF(l3 + \, 0-y-2\+l; 0-a+l: l/.r) + e-2"(+A) j^where the upper or lower sign is taken according as / () 5 0, and it is supposed that I arg a;

j$ tt.

In order to employ the method of steepest descents we have to determine the stationarypoints of (1 wt) t~^(l ty- qua function of f ; they are given by the roots of the quadratic

2wt"- - 3i + 1 = 0.

Put 9 8ii' = z, it being understood that | arg z' Ktt [the cut from x =1 to x = + x in thea;-plane insures this inequality being satisfied if we define arg (9 8.1;) to be zero when .1: = 0],and the stationaiy points are

t, = 2l(S+^z). t,= 2l{S-^z).The vahies of (1 .rf)"' (1 t)~- a.t ti, L are respectively

1 Wz + 3)V(^/^ + 1), i Wz - Sf/Wz - 1).We shall now discuss, by electrical methods*, the topography of the contours in the f-plane

(for all assigned values of z) which are supplied by the application of the method of steepestdescents.

If we write TTT^-^=''*' '^'

where V and W are real, it is evident that V is the potential at the point t due to a two-dimensional electrical distribution consisting of line charges through the points 0, 1, l!x in the

0, to secure the con-vergence of /j and /) which are required by the method of steepest descents. We can nowconsider the topography of the ditierent equipotentials obtained by varying V from + x to x .

* I should have preferred lo have employed the algebraic (in general) only two nodes, they are not unicursal curves,methods of Part I in discussing the forms of the contours and so the algebra appeared intraetible. The investigationinstead of this combination of geometrical and electrical actually given is, I think, quite rigorous,theories ; but as the contours are portions of quartics with

392

/

304 Dr WATSON. ASYMPTOTIC EXPANSIONS

When V is large and positive, the equipotential consists of two small ovals* surroundingthe positive charges at 0, I respectively. As V decreases, the ovals increase in size, until wereach an equipotential through that one of the equilibrium points at which tlie potential is

higher ; this equipotential has a node which may arise in one or other of two ways, (I) by thetwo ovals uniting to form a figure-of-eight or (II) by previously distinct parts of the same

oval bending round towards one another and uniting. Case (I) is shewn in Fig. 8 and casef

(II) in Fig. 9.

Fig. 8.

Fio. 9.

First take case (I). As V decreases further the equipotential becomes a single oval sur-rounding the figure-of-eight and this form persists until wc reach the equipotential through theequilibrium point with lower potential; the node at the equilibrium point can only be formed

by distinct parts nf the oval uniting to surround a portion of the plane not previouslyenclosed ; this area having a portion of an equipotential as its complete boundary must containa charge ; this can only be the charge at t^ = l/x. .Subse(]Uont equipotentials consist of two

ovals, a large one surrounding all three charges and a small one inside the former surroundingthe charge at


Next we take case (II). As V decreases further, the equipotentials become tripartite,consisting of an oval round the charge at 0, another round the charges at 1 and ^3, and a third,inside the second, round the charge at ^ only. This form persists until the first two unite atthe remaining equilibrium-point to form a figure-of-eight ; and subsequent equipotentials arebipartite, consisting of a large oval round all three charges and a smaller one round the chargeat t, only.

If now we regard a; (and therefore z) as a variable, the transition fi-om case (I) to case (II)can only occur when 2 passes through a value which makes the nodal equipotentials coincident

;

i.e. when 2 satisfies the equationWz - sykIz-I

The curve in the ./-plane on which this equation is satisfied is shewn* in Fig. 10; thesimplest form of the equation of the curve is obtained by writing z = re'* (? ^ 0, it ^6 ^ ir),when the equation of the curve reduces to

r = 6V3cosi0-9,together with the coincident rays cos \d=Q {i.e. = it).

/

Fig. 10.

It is easy to see that when co is outside the curve of Fig. 10 and fairly near the oiigin (sothat >Jz is comparatively nearly equal to 3), the potential at t^ is higher than that at t^. And,when ja!| is very small the charge at t3{= 1/x) has little influence on the form of the equi-potentials moderately' near and 1 ; and so the equipotentials moderately near and 1 havenearly the form which thej^ would have if the only existing charges were at and 1. Hence,whenx is outside the curve of Fig. 10, the equipotentials have the configuration of case (I); and thenode of the figure-of-eight is at t^, while the node of the other nodal equipotential (which may bedescribed as a closed crescent) is at t^.

When \x\\ is small, so arei

^3 - 1 and \z\\\ and, if we consider the special case inwhich ^3 1 is positivet, the equilibrium point t, is on the right of t^ and the potential there,viz. log {^ (3 >Jzyi{-Jz l)j , is much higher than the potential at ^, and

306 Dr WATSON, ASYMPTOTIf EXPANSIONS

of Fig. 9 whenever x is inside* the curve of Fig. 10. It is to be noted that ti is always the node

of the figure-of-eight.

[As a confirmation of these results, take \ x i large, when the equilibrium points are both near theorigin ; when we take z positive, t^ is nearer the origin than ?.,, and the potential of t^ is higher thanthat of 0, tt^^^tt); the difference is an oddmultiple of tt on the branch / = 9 -I- 6 V3 1 sin |^ | near z =9, and so the difference is an evenniultiple of TT only when sin If^ = or 7-= 9 fi V3 '^in i^ ; this curve is shewn by a dottedline in Fig. 10, and it lies wholly oidside the curve r = 6v'3 cos J ^ 9. It follows that, as ./

varies inside the continuous curve of Fig. 10, the general configuration of the branch of the lino

of force fi-om to 1 through t, does not change, but always lies inside the figure-of-eight and

* And so no case arises in which the closed-orescent justifies the statement made in footnote +, p. 304.equipotential contains the charges at 0, 1/x only; thia t Whether / (x) be positive or negative.


does not go near* ?.. Also, as we may suppose that t^ and ^3 do not cross the real axis, and asthe branch of the line of force is reconcilable with the real axis, so far as t, is concerned, when/ (1 )

I

is very small and iJ (1 a.) is small and positive, it follows that, for all positions of xinside the continuous curve of Fig. 10, the branch of the line of force under consideration isreconcilable with the real axis.

It is now a simple matter to apply the method of steepest descents to obtain the asymptoticexpansion of/,.

Writing ( 1 - xt) r' (1 - t)-' = 1 [Wz + 8f Wz + l ){ e',where t is po.sitive, we get

and dtldT is expansible (near t = 0) in a series of ascending powers of t^ commencing with aterm in t ~ - whose coefficient is

and hence, after the manner of Part I, we obtain an asymptotic expansion for I^, in descendingpowers of X, of which the dominant term is given by the formula

This fornmla is valid for a complete range of values of argX provided that R{t) is negativewhen t = t.,; i.e. provided that the potential of t^ is higher than the potential of t,,. Consequentlythe formula is valid for a complete or for an incomplete range of values of arg X according asX is outside or inside the continuous curve of Fig. 10.

We shall finally shew that a branch of the line offorce through t, either starts from 1, encirclestj, and returns to 1 ; or else it starts from 1 and ends at O. The former is the case when x isinside the dotted curve of Fig. 10, and the latter when x is outside it.

First suppose that x is inside the continuous curve of Fig. 10 ; then the equipotentials havethe configuration of case (II). Consider the branch of the line of force which enters the hornsof the closed crescent at i,; it cannot cross the boundaries of the crescent without passingthrough an equilibrium point, and no such point exists ; hence both ends of the branch mustterminate at the point 1. Now the configuration of the lines of force only alters when x crossesthe dotted curve of Fig. 10. Hence, whenever x is inside the dotted curve of Fig. 10, there is abranch of a line of force which starts from 1, goes to t^, and returns to 1, obviously encircling t,,which is in the region surrounded by the crescent.

When X is outside the continuous curve of Fig. 10, the closed crescent contains the pointas well as 1 ; and the only possible change of configuration of the line of force is tb '>ne of itsendsf should be at instead of both being at 1.

^ and of Clask. ^* It can only go near (3 by assuming a form in wliieh it sideration is zero, ami since the ch9

passes through (2, and we have just seen that it cannot assume curve, it would have also to containthis form when x is inside the continuous curve of Fig. 10. at (these charges being numeric

,>,,,, journal ot Mathe-t Both ends cannot beat 0; for suppose the line charges in sign). Therefore the quartic t22i_249

replaced by surface distributions on circular cylinders of forms part would have a cusp at ., Carneaie Institution ofvery small radius; smce, by Gauss' theorem, the total charge it from 0, it would bifurcate befo)inside the (closed) branch of the line of force under con- neither of these events actually

308 Db WATSON, ASYMPTOTIC EXPANSIONS OF HYPERGE( )METRIC FUNCTIONS.

To see that the branch of the line of force has one end at and the other at 1, when x is

outside the dotted curve of Fig. 10, take .r ' large (greater than 9/8 is sufficient) and consider

the Hmiting case when x is positive, so that args is +*7r. In this case the nodal eqiiipotentials

coincide and form a curve consisting of two ovals crossing one another at U , L (which are con-

jugate complexes); the left-hand oval contains the charges at and Yjx, and the right-handoval contains the charges at l/.-r and 1. And obviously the branch of the line of force through Uhas its ends at and 1 ; hence, whenever R (.r - 9/8) is positive and I {a:) is very small, the lineof force must pass very near ; and so it must actually have its end at 0, in \-iew of the manner

in which lines of force radiate from the charge at 0.

Hence, whenever x is outside the dotted curve of Fig. 10, a branch of a Hne of force passes

from to 1 by way of f, ; moreover t, lies in the region between this curve and the real axis ;for U is in the region surrounded by the closed crescent, and is consequently inside the region

bounded by the line of force and any curve joining and 1 and lying wholly inside the crescent

;

and, since L and 4 are on the same side of the real axis, the ciu-ve just mentioned is reconcilable

^vith the real axis so far as fs is concerned. Hence, when x is outside the dotted curve of Fig. 1 0,we get

and it is easy to shew that \jz. 1 >Jz, 3 V- have to be taken to have their arguments

numerically less than tt.

If, however, x is inside the dotted curve of Fig. 10, the function which possesses the asymp-

totic expansion of which the dominant term has just been written down is

I

^3+^-' (1 - t)y-^+'^'' ( 1 - xt)-'->' dt,I

where the contour is described counter-clockwise or clockwise according as /(.r)20.

By writing

asymptotic expansions of hypergeometnc functions

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